CHAPTER GENERAL MATHEMATICS & GEOMETRY

Transcription

1 CHAPTER 13 GENERAL MATHEMATICS & GEOMETRY 13.1 Introduction 13.2 Glossary 13.3 Basic Mathematics 13.4 Geometry 13.5 Metalworking Math Applications 13.6 Unit Measurement & Conversion 13.7 Review Questions 13.1 Introduction This review is organized into the following sections: 1) Basic Mathematics, 2) Geometry, 3) Metalworking Math Applications and 4) Unit Measurement and Conversions. It is intended to be a brief overview of each area. Additional, independent study in each area is recommended to properly prepare for the certification exam. September 2013 (S.G.09.13) Page 1 (General Math & Geometry) Chapter 13

2 13.2 Glossary Acute angle - Any angle that is less than Arc - A continuous portion (as of a circle or ellipse) of a curved line. Chord - A straight line joining two points on a curve. It does not pass through the center of the circle. Circumference - The perimeter or distance around the circle. Concentric circles - Two circles that share the same center point. Cosine - The cosine of an angle equals the adjacent side divided by the hypotenuse. Cotangent - The cotangent of an angle equals the adjacent side divided by the opposite side. Decimal - Decimals are numbers that are expressed in tens or multiples of tens. They are used to express parts of a whole number. Equilateral triangle - A triangle that has three equal sides and three equal angles. Equation - An equation is a statement of equality between two expressions, as 4x = 234. The unknown quantity in an equation is generally designated by the letter x. Isosceles triangle - A triangle that has two equal angles and two equal sides. Mil - A unit of measurement equal to 1/1000 of an inch. Minute - An angular measurement that is 1/60 part of a degree. Obtuse angle - Any angle that is more than 90 0 but less that Percentage - A part of a whole expressed in hundredths. Pitch - The distance between one point on a gear tooth and the corresponding point on the next tooth. The distance between any point on the thread of screw to the corresponding point on an adjacent thread measured parallel to the axis. Proportion - A proportion is the relationship between two ratios, which are equal in value. September 2013 (S.G.09.13) Page 2 (General Math & Geometry) Chapter 13

3 Ratio - The ratio between two quantities is the quotient obtained by dividing the first quantity by the second. For example, the ratio between 12 and 3 is 4. Ratio is generally indicated by the sign (:); thus 12:3 indicates the ratio of 12 to 3. Reciprocal - The reciprocal or inverse ratio is the reciprocal of the original ratio. For example, the reciprocal ratio of 5:7 is 7:5. Right triangle - A triangle that has a 90 0 (right) angle. Scalene triangle - A triangle that has no equal sides or angles. Second - An angular measurement that is 1/60 part of a minute (reference "minute") or 1/3,600 of 1 degree of arc. Sector - A geometrical figure bounded by two radii and the included arc of a circle. Sine - The sine of an angle equals the opposite side divided by the hypotenuse. Square of a number - The product of that number multiplied by itself. For example, the square of 9 is 9 x 9 = 81. The square of a number is indicated by the exponent ( 2 ), thus 9 2 = 9 x 9 = 81. Thou - A shop term indicating a unit of measurement equal to 1/1000 of an inch. Square root - The number which when multiplied by itself, will give a product equal to the given number. For example, the square root of 16 (written 16 ) equals 4, because 4 x 4 = 16. Tangent - The tangent of an angle equals the opposite side divided by the adjacent side. Tangent line - A line that touches a circle on the circumference at a point called the point of tangency. September 2013 (S.G.09.13) Page 3 (General Math & Geometry) Chapter 13

4 13.3 Basic Mathematics Shop mathematics and geometry are necessary skills for the designer, machine operator, inspector and sales engineer. High technology equipment is the norm in today s manufacturing shops. It has relieved many painstaking and tedious shop procedures of past years, however, there is still a need for proper application of shop math in machine inspection, blueprint reading, NC programming, workpiece setup and final workpiece qualification. As a machine tool sales engineer you will become involved in each of these areas in the sale and application of equipment. This general review of shop mathematics should be supplemented with your own selfstudy based on a good shop mathematics and geometry primer. Refer to the "Reference Books & Suggested Reading" section at the end of Volume II. TYPES OF NUMBERS The common types of numbers in machine tool technology fall into several categorieswhole numbers, fractions and decimals. Whole numbers and decimals can be expressed in either the inch or metric system. MATHEMATICAL OPERATIONS In the shop environment solutions require a diversity of math operations. These operations cover the range of basic mathematics, geometry, trigonometry, algebra, statistics and conversions. Typically, knowledge of the following functions is required: Basic Math Functions addition & subtraction multiplication & division square & square root ratios & proportions Geometry & Trigonometry Functions angles, chords & arcs surface area & volume radius, diameter & circumference sine, cosine, tangent, cotangent, etc. September 2013 (S.G.09.13) Page 4 (General Math & Geometry) Chapter 13

5 Typical areas where math may be encountered: Workpiece Functions part dimensions part geometry part accuracy & tolerances part weight & thermal expansion screw pitch conversions - unit & inch/metric Metalworking Functions cutting speed spindle speed pulley ratios & speeds gear ratios horsepower, torque & force cycle times NOTES September 2013 (S.G.09.13) Page 5 (General Math & Geometry) Chapter 13

6 13.4 Geometry BASIC TWO-DIMENSIONAL GEOMETRIC SHAPES Circle Square Rectangle Right Obtuse Trapezoid Parallelogram Triangle Triangle The above shapes may be encountered on any given two-dimensional surface of workpieces. However, since workpieces are three dimensional in nature, there are often three-dimensional geometric shapes that require mathematical solutions. Threedimensional shapes are cubes, prisms, pyramids, wedges, cylinders, cones, spheres, ellipsoids, barrels, paraboloids and so forth. Calculating the pitch of a screw is a good example of mathematics involving three-dimensional geometry. BASIC GEOMETRIC FORMULAS Circle: Area of Circle = A = r 2 Circumference of Circle = C = 2 r = d Diameter of Circle = d = C/ Radius of Circle = r = C/2 Length of Arc = L = r x x angle, in deg. Radius (r) Diameter (d) Sector Chord Arc Circumference (C) Area (A) Square, Rectangle & Right Triangle: Area of Square = A = a 2 Area of Rectangle = A = a x b Area of Right Triangle = A = a x b 2 Diagonal of Square = d = 2a 2 Diagonal of Rectangle or Right Triangle = d = a 2+ b 2 d b d = hypotenuse a September 2013 (S.G.09.13) Page 6 (General Math & Geometry) Chapter 13

7 Circumscribed Squares: Diameter of Circumscribed Circle = d = 2s 2 Side of Square = s= d 2 2 or, s =. d A S=side d B OTHER GEOMETRIC SHAPES & TRIGONOMETRY FUNCTIONS When solving problems involving more complex geometric shapes, the first step is to break those shapes into smaller more common pieces, solve the pieces, and then sum the parts. For example, a parallelogram can be broken down into a rectangle and two triangles. Also, if a triangle is not a right triangle it can be divided into two right triangles. When solving for angles and sides of right triangles you can use trigonometric functions such as sine, cosine, tangent, cotangent, secant and cosecant. You should familiarize yourself with these functions and their basic application. Machinery handbooks contain trig tables, which simplify solving problems involving these functions. In the diagram above, the angles are indicated with capital letters A, B and C. The sides are indicated with small letters a, b and c which are opposite the angle with the same letter. The side opposite the right angle is always the hypotenuse. In the example, the angle A will have the following trigonometric definitions. sine A = side opposite = hypotenuse a c cosine A = side adjacent = hypotenuse c b tangent A = side opposite = side adjacent b a cotangent A = side adjacent = side opposite a b secant A = hypotenuse = side adjacent b c cosecant A = hypotenuse = side opposite a c Side Opposite B a C Hypotenuse c b Side adjacent A September 2013 (S.G.09.13) Page 7 (General Math & Geometry) Chapter 13

8 13.5 Metalworking Math Applications The application of math skills in the metalworking environment covers a wide range. Some of the more important uses are: 1. Calculation of workpiece metalcutting parameters. a. Cutting speed - rpm of tool or workpiece. b. Diameter of cutter (milling, drilling or boring) or workpiece (turning). c. Feedrate or surface speed per minute. d. Surface finish. e. Tooling angles. f. Coefficient of expansion (workpiece or machine). g. Thread pitch. h. Horsepower requirements 2. Calculation of workpiece metalforming parameters. a. Drawing, blanking and punching forces. b. Stock requirements. 3. Calculation of machine parameters. a. Pulley and gear speed ratios. 4. Calculation of Workpiece inspection tolerances. a. Six sigma and Standard deviation. Some of the more common formulas are shown below; however, the machine tool sales engineer will need a working knowledge of the entire math applications indicated above. Cutting Speed: The distance that a point on the circumference of a tool, or on the surface of a workpiece, will travel in one minute. (ft/min or m/min) For inch units only: For metric units only: V = S D V = S D Where: V = Cutter speed in ft/min or m/min S = Spindle speed in rpm D = Diameter (outside) of cutter or workpiece in inches or mm Note: Recommended cutting speed ranges for various materials can be found in metalworking reference books. These ranges are a good beginning point prior to optimization. September 2013 (S.G.09.13) Page 8 (General Math & Geometry) Chapter 13

9 Spindle Speed: The speed at which the cutting tool spindle or workholding spindle rotates (rpm). For inch units only: For metric units only: S = 12V S = 1000V D D Where: S = Spindle speed in rpm V = Cutter speed in ft/min or m/min D = Diameter (outside) of cutter or workpiece in inches or mm Feedrate (milling): The rate of speed that the tool traverses across the workpiece. (in/rev, mm/rev, in/min or mm/min) F = SNC Where: F = Feed, in inches or mm per min. S = Spindle speed in rpm N = Number of cutter teeth C = Chip thickness Note: Recommended chip thickness can be obtained from a metalworking handbook based on the workpiece material. Pulley Speed Or Size: The calculation of pulley sizes and speeds involves the mathematical application of ratios and inverse proportions. Here is the inverse proportion for two pulleys (pulley A and pulley B), each of a different diameter. Diameter Pulley A = Speed of Pulley B Diameter Pulley B Speed of Pulley A If pulley A is 4" diameter and pulley B is 2" diameter and pulley B is rotating at 500 rpm solve for the speed of pulley A. 4 = X X = 2 x 500 = 250 rpm 4 A B 4" 2" 500 rpm. September 2013 (S.G.09.13) Page 9 (General Math & Geometry) Chapter 13

10 13.6 Unit Measurement & Conversion Every quantity, whether it be length, speed, mass, volume, power, etc., is expressed in some standard unit of measure. As a point of interest, until 1960 the unit measure of "one inch" was defined as: "The distance between two marks on a known length standard maintained under conditions of constant temperature and humidity." In 1960 the definition was redefined as: "41, wave lengths on a vacuum of the reddish-orange radiation corresponding to the transition between levels 2p10-5d5 of the unperturbed atom of Krypton." This is a great example of how technology has impacted unit measurement. This technology has allowed greater access to standards in the field, outside the laboratory. The machine tool sales engineer should be familiar with both the inch and metric (International System - SI) units of measure as well as common conversion formulas between them. Handy conversion formulas and charts are available in the reference section of most machining handbooks. However, relationships between such common units as length (meters/inch), mass (pound/gram) and power (hp/watt) should be committed to memory for quick reference when customers present a workpiece drawing. With the push for global manufacturing capabilities, companies are rapidly converting to the metric system. Inch Metric (SI) Length foot, inch meter Mass pound, ounce kilogram Temperature degree Fahrenheit degree Kelvin Force pound-force Newton Energy Btu joule Power horsepower watt Volume gallon liter One of the benefits of the metric system is that most units are based on decimal multiples or decimal sub-multiples of the base standard unit. Since units of different sizes are created by multiplying or dividing the base unit by powers of 10 the measurement process is simplified. Each of these multiples has been assigned a common prefix that allows for continuity across the system. These prefixes are defined as follows: September 2013 (S.G.09.13) Page 10 (General Math & Geometry) Chapter 13

11 Prefix Relationship Multiply Base Example Unit By mega multiplied by one million 1,000,000 megameter kilo multiplied by one thousand 1,000 kilometer hecto multiplied by one hundred 100 hectometer deka multiplied by ten 10 dekameter BASE UNIT 1 METER deci one tenth part.1 decimeter centi one hundredth part.01 centimeter milli one thousandth part.001 millimeter micro one millionth part micrometer (micron) Hear is a brief inch - metric conversion chart for commonly needed dimensions. Inch to Metric Metric to Inch 1.0" = 25.4mm 1000mm = 39.37" 0.1" = 2.54mm 500mm = 19.69" 0.01" =.254mm 100mm = 3.937" 0.001" =.0254mm 10mm = " " = mm 1mm = " 0.1mm = " 0.01mm = " 0.001mm = " The inch system does not have the simplicity of the metric prefix, which is based on the powers of ten. However, there are a few inch terms such as "mil" and "thou" that should be familiar to the sales engineer due to common use in the industry. Refer to the glossary section for definition of these terms. Here's a conversion table for quick reference. a thousandth = 0.001" = mm a tenth (mil) = " = mm a micron = 0.001mm = " or about 40 millionths of an inch September 2013 (S.G.09.13) Page 11 (General Math & Geometry) Chapter 13

12 13.7 Review Questions These review questions are provided for study purposes only and are not on the CMTSE certification exam. Correctly answering these questions does not guarantee a passing exam grade. 1. The partpiece is a one-inch diameter bar. The tooling dictates a 75 per minute (fpm) cutting speed. What RPM (revolutions per minute) will the machine's spindle need to be turning? What is the circumference of a circle having a radius of 4 inches? What is the surface area, in square inches, of a rectangle having sides of 5.5 inches and 6.4 inches? What is the surface speed (feet per minute) of a 4-inch milling cutter that is turning at 1,000 rpm? rpm rpm 3. 1,050 rpm 4. 1,550 rpm 5. If a part is 10.5 inches long, approximately how long is it in millimeters? 1. 27mm mm mm mm 6. A 9-inch diameter pulley is spinning at 350 rpm. What is the diameter of the pulley it is driving at 168 rpm? inches inches inches inches September 2013 (S.G.09.13) Page 12 (General Math & Geometry) Chapter 13

13 7. How many microns are in a meter? 1. One thousand 2. One hundred thousand 3. One million 4. Ten million 8. One kilowatt is approximately the same as: 1. 10,000 kg moved 1 inch in 1 second 2. 1,000 Joules per hour horsepower 4. 4,000 foot-pounds per second 9. One second of an arc is: 1. 1/1,000,000 of 1 degree. 2. 1/360 of 1 degree. 3. 1/3,600 of 1 degree. 4. 1/100,000 of a complete circle. 10. A tool is on sale for $ It has been reduced by 20%. What was the original price? 1. $ $ $ $ September 2013 (S.G.09.13) Page 13 (General Math & Geometry) Chapter 13

14 ANSWERS TO REVIEW QUESTIONS 1. (3) 2. (4) 3. (1) 4. (3) 5. (3) 6. (2) 7. (3) 8. (3) 9. (3) 10. (2) September 2013 (S.G.09.13) Page 14 (General Math & Geometry) Chapter 13